C star algebra

A possibly nonunital complex Banach algebra is a complex vector space $A$ equipped with an associative complex-bilinear multiplication $A\times A\to A$ (A bounded bilinear map between normed spaces for the bilinearity convention) and a norm under which $A$ is a Banach space (Banach space, Real and complex scalar conventions for normed spaces), such that the norm is submultiplicative:

$\mathrm{\parallel }ab\mathrm{\parallel }\le \mathrm{\parallel }a\mathrm{\parallel }\mathrm{\parallel }b\mathrm{\parallel }(a,b\in A)\mathrm{.}$

No multiplicative identity is assumed, and no real-algebra or unital convention is imported from elsewhere.

A complex C*-algebra is a possibly nonunital complex Banach algebra $A$ equipped with an involution $A\to A$, $a\mapsto a^{\ast }$, which is conjugate-linear (Real and imaginary parts, complex conjugation, and modulus) and satisfies

$(a^{\ast }{)}^{\ast }=a,(ab{)}^{\ast }=b^{\ast }{a}^{\ast },\mathrm{\parallel }{a}^{\ast }a\mathrm{\parallel }=\mathrm{\parallel }a{\mathrm{\parallel }}^2(a,b\in A)\mathrm{.}$

The last identity is the C*-identity. Two immediate consequences are worth recording, and both are proved from the displayed axioms alone:

• the involution is isometric: $\mathrm{\parallel }{a}^{\ast }\mathrm{\parallel }=\mathrm{\parallel }a\mathrm{\parallel }$. Indeed $\mathrm{\parallel }a{\mathrm{\parallel }}^2=\mathrm{\parallel }{a}^{\ast }a\mathrm{\parallel }\le \mathrm{\parallel }{a}^{\ast }\mathrm{\parallel }\mathrm{\parallel }a\mathrm{\parallel }$ gives $\mathrm{\parallel }a\mathrm{\parallel }\le \mathrm{\parallel }{a}^{\ast }\mathrm{\parallel }$ when $a\mathrm{\ne }0$, and applying this inequality to $a^{\ast }$ and using $(a^{\ast }{)}^{\ast }=a$ gives $\mathrm{\parallel }{a}^{\ast }\mathrm{\parallel }\le \mathrm{\parallel }a\mathrm{\parallel }$; the case $a=0$ is trivial;
• no norm-uniqueness assertion is part of this definition: such a theorem needs additional hypotheses and proof, and does not follow merely by naming the displayed C*-identity.

Let $A$ and $B$ be complex C*-algebras. A bounded star-homomorphism, or bounded $\ast$-homomorphism, is a bounded complex-linear map $\phi :A\to B$ (A bounded linear operator between normed spaces) satisfying

$\phi (ab)=\phi (a)\phi (b),\phi (a^{\ast })=\phi (a{)}^{\ast }(a,b\in A)\mathrm{.}$

A star-homomorphism is not required to be unital, and it is not required that a unit be present or preserved; when $A$ and $B$ both happen to be unital, $\phi$ is called unital if $\phi (1_A)=1_B$. The bounded nonunital star-homomorphisms are the arrows of the locally compact duality theorem on this page (Locally compact Gelfand duality), with properness imposed there through Approximate unit and proper C star morphism.

• The zero algebra. $A=\{0\}$ is a C*-algebra in this sense; it is not unital in the above convention, since its only element equals both candidate identities and the unital definition requires $1\mathrm{\ne }0$. The zero algebra is treated separately in the representation theorems, where it corresponds to the empty space.
• Isometry of the involution is a theorem, not an axiom. It is derived above from the C*-identity, and is used whenever an estimate for $a^{\ast }$ is needed without an inner product or a Hilbert-space adjoint.
• A bounded star-homomorphism is automatically contractive, and injective one is isometric; neither statement is used as a definition, and neither is proved here.