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

A C-star algebra is a complex Banach algebra with an involution satisfying $\Vert a^{\ast }a\Vert =\Vert a{\Vert }^2$.

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.

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.

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 $u^{\ast }u=u{u}^{\ast }=1$.

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 bounded linear functional $\tau$ on a C-star algebra is positive when $\tau (x)\ge 0$ for every Positive element of a C-star algebra. Positivity impliesand $\Vert \tau \Vert =\tau (1)$; conversely, $\Vert \tau \Vert =\tau (1)$ implies positivity.

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 is a positive functional of norm one. The state space $S(A)$ is a weak-star compact convex subset of $A^{\ast }$.

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.

On the linear span of $1,x$, define $g(a1+bx)=a+b\lambda$. Affine spectral mapping gives $|a+b\lambda |\le \Vert a1+bx\Vert$, making this well-defined with norm one and $g(1)=1$. A norm-preserving Hahn-Banach theorem extension to the whole C-star algebra remains unital; the norm characterization of a positive functional on a C-star algebra makes it a state on a C-star algebra attaining the spectral value.

A pure state is an extreme point of the state space. The Krein-Milman theorem guarantees pure states, and every positive element attains its norm at some pure state.

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.

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.

The absolute value of a linear operator $L$ is the positive square root of an operator $L^{†}L$. Its eigenvalues are the singular values of $L$, its trace is the trace norm of $L$, and it appears in the polar decomposition of a bounded operator $L=U|L|$.

If $T_n=P_{n}A{P}_n^{\ast }$ is a square finite compression with singular value decomposition $T_n=V_n{\Sigma }_n{W}_n^{\ast }$, then $U_n=V_n{W}_n^{\ast }$ is a unitary extension of its polar factor. When $A$ is unitary and $P_n^{\ast }{P}_n\to I$ strongly, $|T_n|\to I$ strongly and hence $U_n{P}_n\to A$ strongly.