Example of *-subalegra in a c* algebra which is not c* algebra
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Answer:
Step-by-step explanation:
The algebra M(n, C) of n × n matrices over C becomes a C*-algebra if we consider matrices as operators on the Euclidean space, Cn, and use the operator norm ||·|| on matrices. The involution is given by the conjugate transpose. More generally, one can consider finite direct sums of matrix algebras. In fact, all C*-algebras that are finite dimensional as vector spaces are of this form, up to isomorphism. The self-adjoint requirement means finite-dimensional C*-algebras are semisimple, from which fact one can deduce the following theorem of Artin–Wedderburn type:
Theorem. A finite-dimensional C*-algebra, A, is canonically isomorphic to a finite direct sum
{\displaystyle A=\bigoplus _{e\in \min A}Ae} A=\bigoplus _{e\in \min A}Ae
where min A is the set of minimal nonzero self-adjoint central projections of A.
Each C*-algebra, Ae, is isomorphic (in a noncanonical way) to the full matrix algebra M(dim(e), C). The finite family indexed on min A given by {dim(e)}e is called the dimension vector of A. This vector uniquely determines the isomorphism class of a finite-dimensional C*-algebra. In the language of K-theory, this vector is the positive cone of the K0 group of A.
A †-algebra (or, more explicitly, a †-closed algebra) is the name occasionally used in physics[5] for a finite-dimensional C*-algebra. The dagger, †, is used in the name because physicists typically use the symbol to denote a Hermitian adjoint, and are often not worried about the subtleties associated with an infinite number of dimensions. (Mathematicians usually use the asterisk, *, to denote the Hermitian adjoint.) †-algebras feature prominently in quantum mechanics, and especially quantum information science.
An immediate generalization of finite dimensional C*-algebras are the approximately finite dimensional C*-algebras.
C*-algebras of operators
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The prototypical example of a C*-algebra is the algebra B(H) of bounded (equivalently continuous) linear operators defined on a complex Hilbert space H; here x* denotes the adjoint operator of the operator x : H → H. In fact, every C*-algebra, A, is *-isomorphic to a norm-closed adjoint closed subalgebra of B(H) for a suitable Hilbert space, H; this is the content of the Gelfand–Naimark theorem.
C*-algebras of compact operators
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Let H be a separable infinite-dimensional Hilbert space. The algebra K(H) of compact operators on H is a norm closed subalgebra of B(H). It is also closed under involution; hence it is a C*-algebra.
Concrete C*-algebras of compact operators admit a characterization similar to Wedderburn's theorem for finite dimensional C*-algebras:
Theorem. If A is a C*-subalgebra of K(H), then there exists Hilbert spaces {Hi}i∈I such that
{\displaystyle A\cong \bigoplus _{i\in I}K(H_{i}),} A\cong \bigoplus _{i\in I}K(H_{i}),
where the (C*-)direct sum consists of elements (Ti) of the Cartesian product Π K(Hi) with ||Ti|| → 0.
Though K(H) does not have an identity element, a sequential approximate identity for K(H) can be developed. To be specific, H is isomorphic to the space of square summable sequences l2; we may assume that H = l2. For each natural number n let Hn be the subspace of sequences of l2 which vanish for indices k ≤ n and let en be the orthogonal projection onto Hn. The sequence {en}n is an approximate identity for K(H).
K(H) is a two-sided closed ideal of B(H). For separable Hilbert spaces, it is the unique ideal. The quotient of B(H) by K(H) is the Calkin algebra