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  "body": "\n## Description\n\nLet $H$ be a complex [[Hilbert space|HM2HHE3J51]] with inner product $\\ip{\\cdot}{\\cdot}$.\nLet $B(H)$ be the set of bounded linear operators $T \\colon H \\to H$, with pointwise addition and scalar multiplication, composition as product, and the [[operator norm|06EKW1AKXF]]\n$$\\norm{T} = \\sup \\set{\\norm{T x} : x \\in H, \\norm{x} \\le 1}.$$\nThe involution is the [[adjoint|6HSJBJ8TTQ]] $T \\mapsto T^*$, defined by $\\ip{T x}{y} = \\ip{x}{T^* y}$ for all $x, y \\in H$.\nWith these operations $B(H)$ is a C*-algebra with unit the identity operator.\n\nCheck: $B(H)$ is complete in the operator norm because $H$ is complete, and $\\norm{S T} \\le \\norm{S} \\norm{T}$, so it is a Banach algebra.\nThe adjoint is conjugate linear, satisfies $(S T)^* = T^* S^*$ and $T^{**} = T$, and $\\norm{T^*} = \\norm{T}$.\nFor the C* identity, $\\norm{T x}^2 = \\ip{T^* T x}{x} \\le \\norm{T^* T} \\norm{x}^2$ gives $\\norm{T}^2 \\le \\norm{T^* T}$, and $\\norm{T^* T} \\le \\norm{T^*} \\norm{T} = \\norm{T}^2$ gives the reverse inequality.\n\n## Remarks\n\nIf $H = \\C^n$, then $B(H)$ is the algebra $M_n(\\C)$ of $n \\times n$ complex matrices, and $T^*$ is the conjugate transpose.\nBy the Gelfand–Naimark theorem every C*-algebra is isometrically $*$-isomorphic to a norm-closed $*$-subalgebra of some $B(H)$.\nThe algebra $B(H)$ is also a von Neumann algebra: it is closed in the weak operator topology, and its commutant consists of the scalar multiples of the identity.\nIf $H$ is infinite-dimensional, the compact operators form a proper closed two-sided ideal of $B(H)$, and when $H$ is separable it is the only one other than $0$ and $B(H)$.\n\n## Explanations\n\n### L1 human\n\nThe bounded operators on a Hilbert space are the infinite-dimensional version of square matrices.\nThey can be added, multiplied by composition, and they have an adjoint, the analogue of the conjugate transpose.\nThe size of an operator is the largest factor by which it stretches a vector.\nMost of operator theory studies single elements, subalgebras or subspaces of this one algebra.\n",
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