C-star algebra ($C^{\ast }$-algebra)

A C-star algebra is a complex Banach algebra with an algebra involution $a\mapsto a^{\ast }$ satisfying the C-star identityThe algebra involution is conjugate-linear, reverses products and obeys $(a^{\ast }{)}^{\ast }=a$. An identity need not be present.^[ref]

Examples include complex matrices with the operator norm and conjugate transpose, bounded linear operators on a Hilbert space with the operator adjoint, and $C_0(X)$ on a Hausdorff space that is a locally compact space with pointwise complex conjugation and the supremum norm. The identity ties the norm to the algebraic product and involution: it implies spectral radius norm equality for normal elements and makes an injective C-star homomorphism an isometry. This rigidity permits continuous functional calculus for normal C-star elements.

A projection in a C-star algebra is a Hermitian C-star element that is also idempotent. Its spectrum is contained in $\{0,1\}$, and a nonzero projection has norm one by the C-star identity. If $u$ is a partial isometry, $u^{\ast }u$ and $u{u}^{\ast }$ are its initial and final projections. For a projection $p$, the closed subalgebra $pAp$ has unit $p$ and the inherited norm and involution; it is the corresponding corner C-star algebra. In an operator representation, such an element acts as an orthogonal projection.

For a nonunital C-star algebra, adjoining an identity gives multiplication $(a,\lambda )(b,\mu )=(ab+\lambda b+\mu a,\lambda \mu )$ and involution $(a,\lambda {)}^{\ast }=(a^{\ast },\overline{\lambda })$. Its standard C-star norm extends that of $A$, with the new identity of norm one. It can be constructed by adjoining the identity operator to a faithful nondegenerate operator representation and taking the operator norm. The original algebra is a closed ideal. The usual spectrum of an element in a nonunital algebra is taken in this unitization, permitting unital spectral arguments without requiring the original algebra to contain an identity.

Evaluation at zero is a bounded homomorphism on $C[0,1]$. Its kernel $I$ is a closed self-adjoint ideal and hence a C-star subalgebra. It has no identity: applying a proposed identity to $f(t)=t$ would force that identity to equal one at every positive point, contradicting continuity and its value zero at zero. The map $(f,\lambda )\mapsto f+\lambda 1$ from the unitization of a C-star algebra is a bijective star homomorphism, with inverse $h\mapsto (h-h(0),h(0))$. Pulling back the supremum norm verifies the C-star identity, so this is the prescribed unitization norm. Thus adjoining the missing constant direction recovers all continuous functions on the interval.

The left multiplication representation of a C-star algebra is isometric: test $L_a$ on $a^{\ast }/\Vert a\Vert$ to get $\Vert L_a\Vert =\Vert a\Vert$. For a nonunital algebra, its closed image plus the identity operator is the algebraic unitization. For a formal $z=a+\lambda 1$ and $b\in A$, $\Vert zb{\Vert }^2=\Vert b^{\ast }{z}^{\ast }zb\Vert \le \Vert b\Vert \Vert L_{z^{\ast }z}b\Vert$. Thus $\Vert L_z{\Vert }^2\le \Vert L_{z^{\ast }z}\Vert \le \Vert L_{z^{\ast }}\Vert \Vert L_z\Vert$; applying the same estimate to $z^{\ast }$ proves the algebra involution is isometric and the C-star identity holds. This gives a unitization without first assuming an operator representation on a Hilbert space.

For an infinite-dimensional complex Hilbert space, the Calkin algebra is the unital quotient of the bounded operators by the closed two-sided ideal of compact operators. Its spectrum records invertibility up to compact errors. An operator is Fredholm exactly when its image in this quotient is invertible.

A C-star subalgebra is a norm-closed subalgebra closed under the involution, with the inherited C-star identity. A unital inclusion in this context uses the same identity as the ambient C-star algebra. The spectral permanence for C-star algebras says the inclusion preserves the spectrum of an element.

A C-star homomorphism is a complex linear multiplicative map between C-star algebras that preserves the involution. Such maps are contractive; an injective one is an isometry.

A C-star isomorphism is a bijective C-star homomorphism: a complex-linear multiplicative map preserving the involution. Between unital C-star algebras it automatically preserves the unit. It preserves the Banach algebra spectrum, since it preserves invertibility in both directions. Applying the C-star identity and the spectral norm formula to $a^{\ast }a$ gives $\Vert \Phi (a){\Vert }^2=\Vert \Phi (a^{\ast }a)\Vert =\Vert a^{\ast }a\Vert =\Vert a{\Vert }^2$, so it is automatically isometric.

An injective C-star homomorphism preserves norms. For unital maps, reduce to the commutative algebra generated by $1$ and $a^{\ast }a$. Its induced map on character spaces has dense, hence full, image; otherwise a nonzero continuous function vanishing on that image would lie in the kernel. The C-star identity then gives norm preservation for $a$.

If $A$ is a closed C-star algebra inside a unital C-star algebra $B$ and the two algebras share their identity, an element of $A$ is invertible in $A$ exactly when it is invertible in $B$. Consequently its spectrum of an element does not depend on which of these two algebras is used.

For a Normal element of a C-star algebra $a$ in a unital C-star algebra, the Commutative Gelfand--Naimark theorem identifies $C(\sigma (a))$ isometrically with the closed unital algebra generated by $a$ and $a^{\ast }$. The coordinate function maps to $a$, defining $f(a)$ for every continuous $f$ on the spectrum of an element.

For a normal C-star element $x$ in a nonunital algebra, take its spectrum of an element in the C-star unitization. Zero belongs to that spectrum. Continuous functions on this compact spectrum that vanish at zero form an ideal canonically identified with the displayed function algebra. Their values under the unital continuous functional calculus belong to the original algebra because the scalar quotient evaluates them at zero. Star-polynomial approximation with the constant term removed gives exactly the image $C^{\ast }(x)$.

For a Positive element of a C-star algebra $x$, the displayed improper integral converges in norm. Near zero its integrand has norm at most $t^{-1/2}$; near infinity it has norm at most $\Vert x\Vert t^{-3/2}$. The scalar identity follows by setting $t=s{u}^2$ for $s>0$ and using ${\int }_0^{\infty }2/(1+u^2)du=\pi$; for $s=0$ both sides vanish. The bounds are uniform on the compact spectrum of an element, so continuous functional calculus gives the operator identity.

For Positive elements of a C-star algebra $x\le y$, inversion reverses the order of strictly positive elements, so $(t1+x{)}^{-1}\ge (t1+y{)}^{-1}$ for $t>0$. Subtracting their square-root resolvent integral in a C-star algebra formulas givesThe integrand is positive, and the positive cone is closed, so the norm limit of these integrals is positive. No commutativity of $x$ and $y$ is required. In particular, $b^2\ge a^2$ for positive $a,b$ implies $b\ge a$, even though squaring itself is not generally order preserving.

For bounded self-adjoint $a_i,a$, no uniform bound on $\Vert a_i\Vert$ is needed. The resolvent identity gives $(a_i-i{)}^{-1}-(a-i{)}^{-1}=(a_i-i{)}^{-1}(a-a_i)(a-i{)}^{-1}$, and the first factor has norm at most one. Thus both resolvents at $i$ and $-i$ converge strongly. Products preserve strong convergence for uniformly bounded nets; uniform approximation by the algebra of these resolvent functions then gives the displayed statement for every continuous function vanishing at infinity.

For a normal operator with disconnected spectrum $K=K_1\bigsqcup K_2$, the indicator function of a nonempty proper clopen component union is continuous on $K$. The continuous functional calculus makes it a nonzero proper self-adjoint linear projection commuting with $T,T^{\ast }$. Its range is therefore a nontrivial closed reducing subspace. No eigenvector is needed.

The C-star identity is the norm axiom $\Vert a^{\ast }a\Vert =\Vert a{\Vert }^2$. It implies that the involution is isometric and that a normal element has norm equal to its spectral radius.

Every commutative unital C-star algebra $A$ is isometrically star-isomorphic to $C({\Phi }_A)$ through its Gelfand transform, where ${\Phi }_A$ is its character space of an algebra. Thus abstract algebra elements can be treated as continuous functions on a compact Hausdorff space.

This is the isometric commutative C-star specialization of the Gelfand representation.

An element $a$ of a C-star algebra is positive when $a=b^{\ast }b$ for some $b$, equivalently when $a=a^{\ast }$ and $\sigma (a)\subseteq [0,\infty )$. Continuous functional calculus gives it a unique positive square root $a^{1/2}$.

The positive elements of a C-star algebra form a norm-closed proper convex cone. For $p,q\ge 0$ and $s=\Vert p\Vert +\Vert q\Vert$, continuous functional calculus gives $\Vert s1-p-q\Vert \le s$, forcing the real spectrum of $p+q$ to be nonnegative. The same bound with one fixed upper norm bound proves norm closedness under limits; scaling by nonnegative real scalars preserves positivity. If both $h$ and $-h$ are positive, their spectra lie in $\{0\}$, and spectral radius norm equality for normal elements forces $h=0$. This cone defines the order $a\le b$ precisely when $b-a\in A_{+}$.

In $M_2(\mathbb{C})$ take $a=\mathrm{diag}(1,2)$ and $b=\left(\begin{array}{cc}2& 1\\ 1& 3\end{array}\right)$. Then $a-1\ge 0$ and $b-a=\left(\begin{array}{cc}1& 1\\ 1& 1\end{array}\right)\ge 0$, but $b^2-a^2=\left(\begin{array}{cc}4& 5\\ 5& 6\end{array}\right)$ has determinant $-1$ and a negative eigenvalue. In contrast, for commuting positive elements $b^2-a^2=(b-a)(b+a)$ is positive in the commutative function algebra supplied by continuous functional calculus. The obstruction is noncommutativity, not a failure of the scalar inequality.

Here strictly positive means positive and invertible. Set $k=a^{-1/2}b{a}^{-1/2}\ge 1$ using congruence preserves positivity in a C-star algebra. The continuous functional calculus gives $k^{-1}\le 1$, since $1-t^{-1}\ge 0$ for $t\ge 1$. Then $a^{-1}-b^{-1}=a^{-1/2}(1-k^{-1})a^{-1/2}\ge 0$. The proof works without commutativity. If $1\le a$, functional calculus also gives $a^{-1}\le 1$.

For a positive element $a$, its positive square root in a C-star algebra gives $c^{\ast }ac=(a^{1/2}c{)}^{\ast }(a^{1/2}c)$, which is positive. Applying this to $b-a$ shows that $a\le b$ implies $c^{\ast }ac\le c^{\ast }bc$. The map need not preserve multiplication, but it preserves order; if $c$ is invertible, the inverse congruence also preserves order.

Define positivity spectrally: a self-adjoint element has nonnegative spectrum. First, addition preserves it. For $p,q\ge 0$ and $s=\Vert p\Vert +\Vert q\Vert$, functional calculus gives $\Vert s1-p-q\Vert \le s$, forcing every real spectral value of $p+q$ to be nonnegative. The cone is proper because a self-adjoint element with spectrum contained in $\{0\}$ has norm zero.

For $h=a^{\ast }a$, let $h_{-}=\max (-h,0)$ by the continuous functional calculus, and set $c=a{h}_{-}^{1/2}$. Then $c^{\ast }c=-h_{-}^2$ is nonpositive. The nonzero spectra of products in opposite orders imply that $c{c}^{\ast }$ is nonpositive too. Write $c=u+iv$ with $u,v$ self-adjoint. Then $c^{\ast }c+c{c}^{\ast }=2(u^2+v^2)$ is positive, since self-adjoint squares have nonnegative spectrum. It is also nonpositive, hence zero. It follows that $c^{\ast }c=-c{c}^{\ast }$ is both positive and nonpositive, hence zero. The C-star identity gives $c=0$, so $h_{-}^2=0$ and $h_{-}=0$. Thus $h\ge 0$, without assuming positivity of adjoint products in an earlier step.

Every Positive element of a C-star algebra $a$ has a unique positive element $a^{1/2}$ satisfying $(a^{1/2}{)}^2=a$. It is obtained by applying the continuous functional calculus to $t\mapsto \sqrt{t}$ on $\sigma (a)\subseteq [0,\infty )$.

An element $a$ of a C-star algebra is positive if and only if it is Hermitian and $\sigma (a)\subseteq [0,\infty )$. This characterization transfers the pointwise order on continuous functions to an abstract C-star algebra through the continuous functional calculus.

An element $x$ of a C-star algebra is Hermitian when $x=x^{\ast }$. Its spectrum is real, its norm equals its spectral radius, and it is the difference of two positive elements.

An element $u$ of a unital C-star algebra is unitary when $u^{\ast }u=u{u}^{\ast }=1$.

For a Unitary element of a C-star algebra $u$, the displayed element is also unitary. Its denominator has a multiplicative inverse by the Neumann series, since $\Vert \overline{\lambda }u\Vert <1$. The spectrum of $u$ lies on the unit circle, where $|t+\lambda |=|1+\overline{\lambda }t|$. The resulting continuous function $f(t)=(t+\lambda )/(1+\overline{\lambda }t)$ therefore has $\overline{f}f=f\overline{f}=1$ on the spectrum. The continuous functional calculus preserves products and the involution, giving $f(u{)}^{\ast }f(u)=f(u)f(u{)}^{\ast }=1$. The complex conjugate in the denominator is necessary for the unit-modulus calculation.

An element $x$ of a C-star algebra is normal when $x^{\ast }x=x{x}^{\ast }$.

Since a normal C-star element commutes with its adjoint, words in those two generators can be rearranged into the displayed finite sum. Including the constant term generates the unital star algebra; omitting it generates the nonunital star algebra. Under continuous functional calculus, such a sum is the ordinary function ${\sum }_{j,k}{c}_{jk}{z}^j{\overline{z}}^k$ on the spectrum. The Stone-Weierstrass theorem makes these functions uniformly dense.

The unital continuous functional calculus has image $C^{\ast }(1,x)$, the norm closure of star polynomials including constants. The smallest closed star subalgebra $C^{\ast }(x)$ need not contain the ambient identity: for $x=0$ it is zero. In a nonunital algebra, apply the unital calculus in the C-star unitization and retain functions vanishing at zero to recover $C^{\ast }(x)$ inside the original algebra.

For normal $x$, the closed unital C-star subalgebra generated by $x,x^{\ast }$ is commutative. When the ambient spectrum is real, its complement is connected. Membership of the resolvent in this subalgebra is both open and closed on that complement, and holds at large modulus by the Neumann series. Thus the subalgebra spectrum is the same real set. Its isometric Gelfand transform takes $x$ to a real-valued function and $x^{\ast }$ to its conjugate, proving equality. Normality is necessary: the nonzero matrix $\left(\begin{array}{cc}0& 1\\ 0& 0\end{array}\right)$ has spectrum $\{0\}$ but is not self-adjoint.

For a Normal element of a C-star algebra, the C-star identity gives $\Vert a^2\Vert =\Vert a{\Vert }^2$, and iteration gives $\Vert a^{2^n}\Vert =\Vert a{\Vert }^{2^n}$. The spectral radius formula then gives $r(a)=\Vert a\Vert$. This proves the isometry of the Gelfand transform for a commutative C-star algebra directly from its defining norm identity.

A complex linear functional $\tau$ on a C-star algebra is positive when $\tau (a)\ge 0$ for every Positive element of a C-star algebra $a$. Such a linear functional is automatically bounded and satisfies the Cauchy–Schwarz inequality for positive C-star functionals:For a unital C-star algebra, positivity gives $\Vert \tau \Vert =\tau (1)$. Conversely, a bounded complex linear functional with $\Vert \tau \Vert =\tau (1)$ is positive. The unit assumption belongs to these formulas; it is not needed for the definition of positivity.

A concrete example on complex matrices is $\tau (a)=\mathrm{tr}(\rho a)$ with $\rho$ a positive semidefinite matrix. It is a state on a C-star algebra when $\mathrm{tr}\rho =1$. The positivity follows from $\mathrm{tr}(\rho b^{\ast }b)=\mathrm{tr}((b{\rho }^{1/2}{)}^{\ast }b{\rho }^{1/2})\ge 0$.

Positivity of $f((x+ty{)}^{\ast }(x+ty))$ for every complex $t$ proves the displayed Cauchy-Schwarz inequality, including zero diagonal cases. Replacing $x,y$ by their adjoints gives $|f(x{y}^{\ast }){|}^2\le f(x{x}^{\ast })f(y{y}^{\ast })$. These two forms cannot be mixed for a general positive functional on a C-star algebra. For the state evaluating the lower-right entry of $2\times 2$ matrices and $x=y=E_{12}$, $f(x^{\ast }y)=1$ but $f(x{x}^{\ast })=f(y{y}^{\ast })=0$, disproving the mixed form.

A positive bounded linear functional is tracial when its value on a product is unchanged by reversing the two factors. When normalized by $\tau (1)=1$, it is a tracial state. A vector functional $\tau (a)=⟨a\Omega ,\Omega ⟩$ is faithful if the vector is separating, since $\tau (a^{\ast }a)=\Vert a\Omega {\Vert }^2$. Its trace identity makes the associated Tomita operator isometric.

A state on a C-star algebra is a positive functional on a C-star algebra of norm one. For a nonzero unital C-star algebra $A$, its state space $S(A)$ is a compact for the weak-star topology convex set: positivity and $f(1)=1$ define a closed for the weak-star topology subset of the dual unit ball, and the Banach-Alaoglu theorem gives compactness.

For a nonunital C-star algebra, the state space need not be compact for the weak-star topology. In $C_0(\mathbb{R})$, evaluations ${\delta }_n(f)=f(n)$ are states, but $f(n)\to 0$ for every $f\in C_0(\mathbb{R})$. Thus ${\delta }_n$ converges in the weak-star topology to the zero functional, which has norm zero and is not a state.

For a nonzero unital C-star algebra, a positive functional on a C-star algebra obeys $\Vert x\Vert =x(1_A)$, and conversely this equality for a bounded functional implies positivity. Therefore its state on a C-star algebra functionals are exactly the maximizers of the real linear evaluation $x\mapsto \mathrm{Re}x(1_A)$ on the closed dual unit ball. If a convex combination reaches that maximum, each constituent must reach it: each real part is at most one, and their weighted mean is one. The modulus bound then forces evaluation to equal one, so each constituent is again a state. Hence the state space is an exposed face and a face of a convex set.

For a unital represented C-star algebra $A\subseteq \mathcal{B}(H)$, the displayed functional is a state. It is linear under the first-argument-linear inner-product convention, has $x_{\eta }(1)=1$, and is positive because $x_{\eta }(a^{\ast }a)=\Vert a\eta {\Vert }^2\ge 0$. The Cauchy-Schwarz inequality gives $|x_{\eta }(a)|\le \Vert a\Vert$, while evaluation at the unit gives equality of its norm with one. Its GNS null left ideal consists exactly of the operators annihilating $\eta$.

For a state on a C-star algebra $x$ on a unital algebra $A$, put $⟨[a],[b]⟩=x(b^{\ast }a)$ on $A/N_x$, with the inner product linear in its first argument. The Cauchy–Schwarz inequality for positive C-star functionals makes the null vectors pair to zero with every vector. Moreover $c^{\ast }c\le \Vert c{\Vert }^{2}1$ gives $x((ca{)}^{\ast }(ca))\le \Vert c{\Vert }^{2}x(a^{\ast }a)$, so $N_x$ is a left ideal and $[a]\mapsto [ca]$ is well-defined and bounded by $\Vert c\Vert$. Completion gives a Hilbert space $H_x$ and a star-representation ${\pi }_x(c)[a]=[ca]$. The adjoint identity follows from $x(b^{\ast }ca)=⟨[a],{\pi }_x(c^{\ast })[b]⟩$. The unit vector ${\xi }_x=[1]$ is cyclic and recovers the state as $x(a)=⟨{\pi }_x(a){\xi }_x,{\xi }_x⟩$.

On the C-star algebra of convergent complex sequences, a functional in the displayed coordinate representation is a state on a C-star algebra exactly when all its weights are nonnegative and sum to one. Coordinate projections test positivity of $w_j$ at finite indices. Positive sequences with the first $N$ terms zero and the rest one test $w_{\infty }$ after passage to the limit. The known dual isometry makes the norm the sum of the absolute weights, proving necessity and sufficiency.

If only finitely many weights are positive, the GNS construction identifies the quotient by $\{\alpha :F({\alpha }^{\ast }\alpha )=0\}$ with the values at precisely those supporting indices. Its inner product is the weighted coordinate inner product. Passing to the standard Hilbert norm gives diagonal evaluation operators and cyclic vector whose entries are the square roots of the weights. Arbitrary finite coordinate values and the limiting value can be assigned independently by a convergent sequence, which proves cyclicity. For weights $1/2$ at infinity and $1/4$ at each of the first two indices, the space is ${\mathbb{C}}^3$ and the cyclic vector is $(1/\sqrt{2},1/2,1/2)$.

For the state on a C-star algebra $a\mapsto \mathrm{Tr}(\rho a)$ on $M_d(\mathbb{C})$, the GNS construction has the displayed Hilbert-space dimension. Here $\rho$ is a density matrix. Its null space for the GNS seminorm consists of $b$ with $b{\rho }^{1/2}=0$, because $\mathrm{Tr}(\rho b^{\ast }b)=\Vert b{\rho }^{1/2}{\Vert }_{\mathrm{HS}}^2$. Thus the GNS space is the space of matrices $b{\rho }^{1/2}$ with the Hilbert-Schmidt inner product. If $\rho$ has rank $r$, diagonalization identifies these matrices with $r$ arbitrary columns, giving dimension $dr$. Left multiplication is the representation and ${\rho }^{1/2}$ is its cyclic vector. In particular a pure rank-one matrix state gives the defining $d$-dimensional representation, while a full-rank state gives a $d^2$-dimensional space.

For a vector state on a C-star algebra, $N=\{a:a\eta =0\}$ and $⟨[a],[b]⟩=⟨a\eta ,b\eta ⟩$. Thus $U_0[a]=a\eta$ is a well-defined linear isometry from the quotient pre-Hilbert space onto $A\eta$. It extends to an isometry from the completed GNS space onto $K=\overline{A\eta }$: its range is complete and therefore closed, and contains the dense subspace $A\eta$. The space $K$ is invariant under both $A$ and its adjoints. On quotient vectors, $U\pi (a)[b]=ab\eta =aU[b]$, proving the displayed unitary equivalence after continuity. The cyclic vector maps to $\eta$.

If $A=\mathcal{B}(H)$, then $A\eta =H$, since the rank-one map $v\mapsto ⟨v,\eta ⟩\zeta$ sends $\eta$ to any chosen $\zeta$. In this case, for the rank-one orthogonal projection $p_{\eta }$, $a\eta =0$ is equivalent to $a{p}_{\eta }=0$, by evaluating at $\eta$ and using $(a{p}_{\eta })v=⟨v,\eta ⟩a\eta$.

Write $x=h+ik$ with both terms Hermitian. At least one is nonzero; its norm equals its spectral radius, so it has a nonzero spectral value. Spectral values attained by C-star states gives a state taking that nonzero value on the chosen term. Both state values on $h,k$ are real, so the corresponding complex value on $x$ cannot vanish. This also detects nonzero quasinilpotent elements, which cannot be separated by looking only at their own spectrum.

Let $A$ be a nonzero unital C-star algebra, $x\in A$, and $\lambda \in {\sigma }_A(x)$. On the linear span of $1,x$, define $g(a1+bx)=a+b\lambda$. The spectral mapping theorem for affine polynomials gives $|a+b\lambda |\le \Vert a1+bx\Vert$, making $g$ well-defined with norm one and $g(1)=1$. A norm-preserving Hahn-Banach theorem extension to $A$ remains unital; the norm characterization of a positive functional on a C-star algebra makes it a state on a C-star algebra attaining $\lambda$.

The unit assumption is necessary for this formulation. The element $x(t)=e^{-t^2}$ of the nonunital algebra $C_0(\mathbb{R})$ has zero in its unitization spectrum, but every state integrates against a probability measure on $\mathbb{R}$ and takes a strictly positive value on $x$. The preceding extension proof can be performed in the C-star unitization; a state obtained there need not restrict to a norm-one functional on the original algebra.

A pure state on a C-star algebra is an extreme point of its state space. For a nonzero unital C-star algebra, the state space is a nonempty compact convex set in the weak-star topology, so the Krein-Milman theorem guarantees pure states.

Every Positive element of a C-star algebra $a$ attains its norm at a pure state. Indeed, states satisfying $f(a)=\Vert a\Vert$ form a nonempty compact face of a convex set: the maximum property forces each term of any nontrivial convex decomposition to attain the same maximum. An extreme point of this face is an extreme point of the whole state space. The argument uses the unital compactness statement, rather than asserting compactness for a general nonunital algebra.

For a normal operator $T$ with spectrum $K$, there is a contractive unital star-homomorphismwhere $L^{\infty }(K)$ denotes the bounded Borel functions. It is obtained by integrating against the projection-valued spectral measure of $T$.

For a bounded Borel measurable function $f$ and the projection-valued measure $E$ of a normal operator, the displayed set is the spectrum of an element of $f(T)$. Outside it, a bounded reciprocal away from a spectral-null set supplies the inverse. Inside it, unit vectors supported by the indicated arbitrarily small spectral sets make $f(T)-\lambda$ fail to be bounded below. The kernel of the calculus consists of functions satisfying $E(|f|>\epsilon )=0$ for every $\epsilon >0$, and its norm is the corresponding essential supremum. Ordinary pointwise range is insufficient: for multiplication by $t$ on $L^2[0,1]$, $1_{\{0\}}(T)=0$ even though the function takes the value $1$.

A bounded normal operator admits a normal square root even when no continuous scalar square root exists on its Banach algebra spectrum. Choose a bounded Borel branch $s(z)=|z{|}^{1/2}\exp (i\mathrm{Arg}(z)/2)$, with $s(0)=0$. The Borel functional calculus for a normal operator gives $S=s(T)$; multiplicativity gives $S^2=T$ and preservation of the adjoint gives $S^{\ast }S=S{S}^{\ast }=|s{|}^2(T)$. Unlike a positive square root, this square root need not be unique.

Normal operators $A_n$ on approximating subspaces converge to $A$ in the functional-calculus sense when $f(A_n)$ converges weakly to $f(A)$, after the natural projection and inclusion maps, for every continuous function $f$ on a common compact spectral set.

An operator $U$ on a Hilbert space is a partial isometry when it is isometric on $(\ker U{)}^{\perp }$. Equivalently, $U^{\ast }U$ is the orthogonal projection onto this initial space.

For a nonzero partial isometry $u$ in a C-star algebra, put $p=u{u}^{\ast }$ and $q=u^{\ast }u$. The closed linear space $pAq$ becomes a unital C-star algebra with the displayed operations and inherited norm, with unit $u$. The map $a\mapsto u^{\ast }a$ is an isometric -isomorphism onto the ordinary corner $qAq$, with inverse $b\mapsto ub$. The identities $u{u}^{\ast }a=a$ and $u^{\ast }ub=b$ verify the inverse maps, while contractivity of both multiplications gives isometry. Products and involutions transfer directly, giving all algebra and C-norm axioms. This construction turns a rectangular operator corner into a square algebra by using the partial isometry as its unit.

Every bounded operator on a Hilbert space has a polar decomposition $T=U|T|$, where $|T|=(T^{\ast }T{)}^{1/2}$ and $U$ is a partial isometry with $\ker U=\ker T$. On $\mathrm{im}|T|$, it is defined by $U(|T|x)=Tx$.

This is the bounded-operator case of polar decomposition.

For a bounded linear operator $L$ on a Hilbert space, its operator absolute value is the positive square root of an operatorIt appears in the polar decomposition of a bounded operator $L=U|L|$. The identity $\Vert |L|x{\Vert }^2=⟨L^{\ast }Lx,x⟩=\Vert Lx{\Vert }^2$ shows that $|L|$ retains the size of the action of $L$, while $U$ supplies its isometric part.

For a compact operator, the nonzero eigenvalues of $|L|$, counted with multiplicity, are the singular values of $L$. When $L$ is trace-class, $\mathrm{tr}|L|=\Vert L{\Vert }_1$ is its trace norm. An arbitrary bounded linear operator need not have a discrete sequence of singular values in this sense or a finite trace norm; those conclusions require their stated hypotheses.

Let $P_n:H\to H_n$ be coisometries onto finite-dimensional Hilbert spaces, with $Q_n=P_n^{\ast }{P}_n\to I$ strongly, and let $A$ be a unitary operator on $H$. The square compression $T_n=P_{n}A{P}_n^{\ast }$ has a singular value decomposition $T_n=V_n{\Sigma }_n{W}_n^{\ast }$. Thus $U_n=V_n{W}_n^{\ast }$ is a unitary operator on $H_n$ extending the polar factor, including a choice on its null space.

Because the codomains vary, the convergence statement in $H$ isIndeed, $P_n^{\ast }{T}_n^{\ast }{T}_n{P}_n=Q_n{A}^{\ast }{Q}_{n}A{Q}_n\to I$ strongly, with uniformly bounded operator norms. Polynomial approximation to the square-root function gives $P_n^{\ast }|T_n|P_n\to I$ strongly. Since $T_n=U_n|T_n|$,Finally, $P_n^{\ast }{T}_n{P}_n=Q_{n}A{Q}_n\to A$ strongly. Including the lift $P_n^{\ast }$ makes these comparisons between operators on the same Hilbert space.